Quantum minimum description of density matrices

2026-10-05

AI Summary

This study addresses the minimum memory cost of compressing multiple copies of a density matrix when its spectrum is known but its eigenbasis is not. The proposed approach integrates irreducible representations of GL(d,ℂ) with quantum information theory, constructing an achievable scheme by generalizing the Werner cloning map and establishing a matching lower bound via Koashi–Imoto incompressibility, with formal verification completed using the Lean proof assistant. This work precisely determines the minimal description length constant for fixed dimensions, revealing its deep connections to universal lossless coding overhead and free entropy. Furthermore, it derives finite trace-distance bounds and provides complete machine-checkable proof certificates.

Abstract

We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to $\mathrm{GL}(d,\mathbb C)$ irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi--Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.

Problem

• quantum compression
• density matrices
• minimum description length
• Schumacher compression
• memory cost

Innovation

• Schumacher compression
• Werner cloning map
• Koashi-Imoto incompressibility
• quantum minimum description length
• Lean certificate